Stress Transfer During Pressure Solution Compression of Rigidly Coupled Axisymmetric Asperities Pressed Against a Flat Semi-Infinite Solid

In a previous work, we developed a numerical model of compression by pressure solution (PS) of a single axisymmetric asperity pressed against a flat semi-infinite solid. The dissolution rate at any point along the contact and at any time t was determined by (1) computing the normal stress distribution from the current shape of the asperity, and (2) solving the diffusion equation inside the fluid-saturated solid-solid interface, including local dissolution source terms corresponding to the stress field previously determined. The change in shape of the asperity during an infinitesimal time interval δt can then be calculated and the entire procedure repeated as many times as desired. Our results showed that, as the contact flattens and grows during PS, the initial elastic deformation is partially relaxed and the stress transferred from the contact center to the edge. Our goal in the present paper is to demonstrate that, among a population of asperities, stress can also be transferred from one contact to another and that the overall compaction rate can be significantly affected by this process. For this purpose we extended our previous numerical model to simulate PS of two rigidly coupled spherical asperities simultaneously pressed against a flat semi-infinite solid. We considered two end-member cases: 1) transfer of stress to a newly created, not initially present contact, 2) transfer of stress between asperities with different sizes. In both cases, stress was transferred from the most stressed asperity to the least, and, the overall PS displacement rate was reduced. Thus, formation of new contacts and heterogeneous distribution of asperity sizes, which are both expected to exist in rough fractures with self-affine aperture or in heterogeneous granular materials with variable grain-packing geometry, may significantly slow down PS creep compaction.     

Numerical investigation of alternative fracture stiffness measures and their respective scaling behaviours

We study the mechanical deformation of fractures under normal stress, via tangent and specific fracture stiffnesses, for different length scales using numerical simulations and analytical insights. First, we revisit an equivalent elastic layer model that leads to two expressions: the tangent stiffness is the sum of an “intrinsic” stiffness and the normal stress, and the specific stiffness is the tangent stiffness divided by the fracture aperture at current stress. Second, we simulate the deformation of rough fractures using a boundary element method where fracture surfaces represented by elastic asperities on an elastic half‐space follow a self‐affine distribution. A large number of statistically identical “parent” fractures are generated, from which sub‐fractures of smaller dimensions are extracted. The self‐affine distribution implies that the stress‐free fracture aperture increases with fracture length with a power law in agreement with the chosen Hurst exponent. All simulated fractures exhibit an increase in the specific stiffness with stress and an average decrease with increase in length consistent with field observations. The simulated specific and tangent stiffnesses are well described by the equivalent layer model provided the “intrinsic” stiffness slightly decreases with fracture length following a power law. By combining numerical simulations and the analytical model, the effect of scale and stress on fracture stiffness measures can be easily separated using the concept of “intrinsic” stiffness. We learn that the primary reason for the variability in specific stiffness with length comes from the fact that the typical aperture of the self‐affine fractures itself scales with the length of the fractures.     

An asperity model to simulate rupture along heterogeneous fault surfaces

A model has been developed to simulate the statistical and mechanical nature of rupture on a heterogeneous strike-slip fault. The model is based on the progressive failure of circular asperities of varying sizes and strengths along a fault plane subjected to a constant far-field shear displacement rate. The basis of the model is a deformation and stress intensity factory solution for a single circular asperity under a unidirectional shear stress. The individual asperities are unified through the fault stiffness and the far-field stress and displacement. During fault deformation asperities can fail and reheal, resulting in changes in the local stresses in the asperities, stress drops, and changes in the stiffness of the fault. Depending on how the stress is redistributed following asperity failure and on the strenghts of the neighboring asperities an earthquake event can be the failure of one or more asperities. Following an earthquake event seismic source parameters such as the stress drop, energy change, and moment magnitude are calculated. Results from the model show a very realistic pattern of earthquake rupture, with reasonable source parameters, the proper magnitude-frequency behavior, and the development of characteristic earthquakes. Also the progression ofb-values in the model gives some insight into the phenomenon of self-organized criticality.     

Seismic attenuation and velocity dispersion in fractured rocks: The role played by fracture contact areas

The presence of fractures in fluid‐saturated porous rocks is usually associated with strong seismic P‐wave attenuation and velocity dispersion. This energy dissipation can be caused by oscillatory wave‐induced fluid pressure diffusion between the fractures and the host rock, an intrinsic attenuation mechanism generally referred to as wave‐induced fluid flow. Geological observations suggest that fracture surfaces are highly irregular at the millimetre and sub‐millimetre scale, which finds its expression in geometrical and mechanical complexities of the contact area between the fracture faces. It is well known that contact areas strongly affect the overall mechanical fracture properties. However, existing models for seismic attenuation and velocity dispersion in fractured rocks neglect this complexity. In this work, we explore the effects of fracture contact areas on seismic P‐wave attenuation and velocity dispersion using oscillatory relaxation simulations based on quasi‐static poroelastic equations. We verify that the geometrical and mechanical details of fracture contact areas have a strong impact on seismic signatures. In addition, our numerical approach allows us to quantify the vertical solid displacement jump across fractures, the key quantity in the linear slip theory. We find that the displacement jump is strongly affected by the geometrical details of the fracture contact area and, due to the oscillatory fluid pressure diffusion process, is complex‐valued and frequency‐dependent. By using laboratory measurements of stress‐induced changes in the fracture contact area, we relate seismic attenuation and dispersion to the effective stress. The corresponding results do indeed indicate that seismic attenuation and phase velocity may constitute useful attributes to constrain the effective stress. Alternatively, knowledge of the effective stress may help to identify the regions in which wave induced fluid flow is expected to be the dominant attenuation mechanism.     

A Study of Permeability Changes Due to Cold Fluid Circulation in Fractured Geothermal Reservoirs

Reservoir behavior due to injection and circulation of cold fluid is studied with a shear displacement model based on the distributed dislocation technique, in a poro‐thermoelastic environment. The approach is applied to a selected volume of Soultz geothermal reservoir at a depth range of 3600 to 3700 m. Permeability enhancement and geothermal potential of Soultz geothermal reservoir are assessed over a stimulation period of 3 months and a fluid circulation period of 14 years. This study—by shedding light onto another source of uncertainty—points toward a special role for the fracture surface asperities in predicting the shear dilation of fractures. It was also observed that thermal stress has a significant impact on changing the reservoir stress field. The effect of thermal stresses on reservoir behavior is more evident over longer circulation term as the rock matrix temperature is significantly lowered. Change in the fracture permeability due to the thermal stresses can also lead to the short circuiting between the injection and production wells which in turn decreases the produced fluid temperature significantly. The effect of thermal stress persists during the whole circulation period as it has significant impact on the continuous increase in the flow rate due to improved permeability over the circulation period. In the current study, taking into account the thermal stress resulted in a decrease of about 7 °C in predicted produced fluid temperature after 14 years of cold fluid circulation; a difference which notably influences the potential prediction of an enhanced geothermal system.     

沁水盆地南部煤层气储层压裂过程数值模拟研究

储层改造是煤层气井提高产能的重要措施,水力压裂是煤层气储层改造的重要方法.为研究煤层气储层压裂过程及其天然裂缝对煤储层压裂时破裂压力的影响,本文以山西沁水盆地南部高煤级煤矿区为研究区,运用有限元数值模拟方法,计算不同地应力条件下、裂缝处于不同位置时煤储层的破裂压力.结果表明:(1)不同类型地应力场对破裂压力的影响不同.对于均匀应力场,破裂压力随着围压的增大而增大,其增幅约为围压的两倍;对于非均匀应力场,当一个水平主应力不变时,破裂压力会随着水平主应力差的增加而减少;(2)如果地应力条件不变,煤储层破裂压力随着天然裂缝与最大水平主应力方向夹角的增加而增加,水平主应力差越大煤储层破裂压力增幅也越大;(3)在有天然裂隙的地层中进行压裂,当天然裂缝的方位不同时压裂裂缝既可能是沿着天然裂缝扩展的裂缝,也可能是压裂过程中产生的新裂缝,因此天然裂缝的方位对破裂压力具有一定的影响.     

Implications of principal stress axes and eigenvalue ratios on critical orientation of fractures. Application to two tectonic regions in Alaska

The principal stress axes and eigenvalue ratios of the stress tensors from two active seismic regions in Alaska (eastern Aleutian Arc and eastern Gulf of Alaska) are computed. These results are obtained using focal mechanisms of earthquakes with magnitudes ranging from 5.5 to 8.2 and show for both regions slightly different compressional states with an orientation of the pressure axis of N26°W for the eastern Gulf of Alaska and N27°W for the east Aleutian Arc. The eigenvalue ratios for the eastern Gulf of Alaska and east Aleutian Arc are 0.84 and 0.99, respectively. The algorithm used in computations allows us to detect some focal mechanisms which are incoherent with respect to the obtained stress tensors. These incoherences are explained in terms of some tectonic features of the region. The orientations of possible pre-existing fractures which would need an unrealistically high maximum shear stress to start a slip on their fault planes are also investigated. These critical orientations depend on a constitutive frictional law as well as on eigenvalue ratios and eigenvectors of the stress tensors. The orientation, which is given in our case by the angle between the normal vector to a fracture and the regional tensional axis, is approximately 59° for the eastern Aleutian Arc. This angle ranges from 49° to 59° for the eastern Gulf of Alaska. It must be pointed out that fractures which need unrealistically high shear stress to start slip on their fault planes are defined by a very narrow band of possible angles between tensional direction and the normal vector to fault plane.     

Mechanisms of Pore Pressure-stress Coupling which Can Adversely Affect Stress Measurements Conducted in Deep Tunnels

— Rock-stress measurements performed in boreholes drilled from a tunnel at a depth of 800 m indicate that the minimum stress has been reduced from pre-excavation levels by drawdown in pore pressure. The disturbance to the ambient stresses extends well beyond two tunnel diameters. A model is developed to explain the inferred strong coupling between the pore pressure and minimum stress in the rock mass around the tunnel. The model includes the effects of poro-elasticity of the intact rock and the stiffness of the fractures. For the field situation under study where the fractures are unusually complaint, both poro-elasticity and fracture compliance could be important contributors to the coupling. Changes in pore pressure are shown to produce reductions in minimum stress that are at least 50% of the drop in pore pressure and probably higher. In more general situations, the poro-elastic component is likely to be the more significant of the two mechanisms and alone accounts for a coupling factor of 50%. In this context, the importance of fractures in crystalline conditions probably lies more in their promotion of penetrative drainage than their contribution to coupling coefficient. The results highlight the importance of establishing the extent and distribution of pore pressure drawdown about galleries from which stress measurements are to be conducted.     

Dynamic faulting under rate-dependent friction

We discuss the effects of rate-dependent friction on the propagation of seismic rupture on active faults. Several physicists using Burridge and Knopoff's box and spring model of faulting have proposed that fault complexity may arise from the spontaneous development of a self-similar stress distribution on the fault plane. If this model proves to be correct, it has important consequences for the origin of the complexity of seismic sources. In order to test these ideas on a more realistic earthquake model, we developed a new boundary integral equation method for studying rupture propagation along an antiplane fault in the presence of nonlinear rate-dependent friction. We study rupture dynamics of models with single and twin asperities. In our models, asperities are places on the fault with a higher value of prestress. Othewise all fault parameters are homogeneous. We show that for models with such asperities, a slip velocity weakening friction leads to the propagation of supersonic healing phases and to the spontaneous arrest of fracture if the prestress outside the asperities is low enough. For models with asperities, we can also observe narrow slip velocity pulses, qualitatively similar to the so-called Heaton pulses observed in some earthquake accelerograms. We also observe a complex distribution of stress after the rupture that depends on details of the initial distribution of asperities and on the details of the friction law.     

Analysis of slip-softening instability on a fault with asperities

Slip-softening instability on a vertical strike-slip fault with asperities has been analysed. The fault strength is uniform in depth, but the strength is nonuniform in the strike direction, i.e., there are asperities on the fault. These asperities and other segments of the fault have the same type of constitutive law but different peak stresses. The material surrounding the fault is represented by elastic plates, of which the top and bottom surfaces are stress-free.We use a finite element method to study the evolution of theoretical displacement, stress and strain field with a growing displacement applied at the remote plate ends. The slip and frictional stress are obtained as part of the solution. We have compared the difference of theoretical displacement, strain field and the distribution of frictional stress on the fault between unstable and stable slip. In addition, we have studied the effect of size and strength of asperities on instability, and the softening behaviour of asperities before instability.We find that (1) the failure of the fault zone may be due to either dynamic instability or rapid quasistable slip. A general characteristic of unstable mode is that slippage, on some parts of asperities increases indefinitely for a small finite increase in remote imposed displacement until, immediately before the unstable slip; (2) the size and peak strength of asperities have a large effect on instability. Reducing the size and peak strength of asperities tends to replace inertially unstable deformation with stable deformation; (3) the location with maximum acceleration during unstable slip, as the plausible nucleating seismic source, is in asperities; (4) the shapes of the changes in theoretical stress and strain at a given location, caused by the nonlinear constitutive property of the fault, are all similar whether instability, happens or not. This fact suggests that the changes of peak type or bend type in crustal deformation are not required for earthquake instability.     

Fractures: Finite-size scaling and multifractals

The distributions of contact area and void space in single fractures in granite rock have been determined experimentally by making metal casts of the void spaces between the fracture surfaces under normal loads. The resulting metal casts on 52 cm diameter core samples show a complex geometry for the flow paths through the fracture. This geometry is analyzed using finite-size scaling. The spanning probabilities and percolation probabilities of the metal casts are calculted as functions of observation scale. Under the highest stresses of 33 MPa and 85 MPa there is a significant size-dependence of the geometric flow properties for observation scales smaller than 2 mm. Based on this data, the macroscopic percolation properties of the extended fracture can be well represented by relatively small core samples, even under normal stresses larger than 33 MPa. The metal casts also have rich multifractal structure that changes with changing stress.     

The fractal geometry of flow paths in natural fractures in rock and the approach to percolation

The distributions of contact areas in single, natural fractures in quartz monzonite (Stripa granite) are found to have fractal dimensions which decrease fromD=2.00 to values nearD=1.96 as stress normal to the fractures is increased from 3 MPa up to 85 MPa. The effect of stress on fluid flow is studied in the same samples. Fluid transport through a fracture depends on two properties of the fracture void space geometry. the void aperture; and the tortuosity of the flow paths, determined through the distribution of contact area. Each of these quantities change under stress and contribute to changes observed in the flow rate. A general flow law is presented which separates these different effects. The effects of tortuosity on flow are largely governed by the proximity of the flow path distribution to a percolation threshold. A fractal model of correlated continuum percolation is presented which quantitatively reproduces the flow path geometries. The fractal dimension in this model is fit to the measured fractal dimensions of the flow systems to determine how far the flow systems are above the percolation threshold.     

Regional flow simulation in fractured aquifers using stress-dependent parameters

A model function relating effective stress to fracture permeability is developed from Hooke's law, implemented in the tensorial form of Darcy's law, and used to evaluate discharge rates and pressure distributions at regional scales. The model takes into account elastic and statistical fracture parameters, and is able to simulate real stress-dependent permeabilities from laboratory to field studies. This modeling approach gains in phenomenology in comparison to the classical ones because the permeability tensors may vary in both strength and principal directions according to effective stresses. Moreover this method allows evaluation of the fracture porosity changes, which are then translated into consolidation of the medium.     

Mechanical analysis of en echelon structure and its significance of controlling earthquakes

The paper deals with the mechanical origin and rules ofen echelon folds and their secondary fractures by means of elastic stability theory and nonlinear fracture criterion. Obtains the quantitative relations amongen echelon angles ofen echelon folds, ratios of boundary stresses anden echelon pitches of shear zone under an action of general boundary forces (tension shear, pure shear or compression shear). As an applied example, the paper researches the displacement field, stress field, distortion energy distribution, state of secondary fractures and energy released by fracturing ofen echelon fold structure developed at the east foot of Taihang Mountain. The results of research show that maximum principal (compressive) stresses, maximum shear stresses, high value area of distortion energy are in the nuclear parts ofen echelon folds. In these parts compressive fractures were easily developed in approximately parallel with fold axis. So it is verified that the secondary fracture ofen echelon folds is a mechanism controlling a strong earthquake.     

Can injection tests reveal the potential for fault movements?

The role of fluids in faulting mechanism and triggering earthquakes is widely accepted. The effective-stress law is the basis for the postulated theories. Using a generalized version of this law, applicable to both continuum and discontinuities, hydromechanical behaviour of a horizontal fracture in a hypothetical fluid-injection problem is investigated. In this problem the increasing intake flow rates, unpredictable by the traditional fluid-flow solutions, brings out another significant aspect of the role of the fluid pressure in rocks. By reducing the magnitude of the compressive effective stresses the fluid pressure causes elastic recovery in fractures. Simple rheologic models are used to demonstrate this fact. Such effects may lead to permeability increases in the rock mass, depending on the magnitude of the fluid pressure. Such variations in permeability, however, are governed by the path dependency of the fracture-deformation response. Therefore, a significant increase in permeability is an indication of comparability of the state of stress and the applied fluid pressure.This index may reveal the potential of hydroactivation of faults, as may arise in the regions of dam reservoirs, underground waste injections, and known faults, for certain ranges of working pressures relevant to each of the above-cited situations. Fluid-injection tests under constant working pressures are suggested as a means revealing the likelihood of movement on the faults.     

The role of the chemical environment in frictional deformation: Stress corrosion cracking and comminution

The roles of chemically assisted crack and fracture propagation and chemically assisted comminution in frictional deformation are evaluated in this study. Double cantilever beam (DCB) crack propagation data are presented which show that the role of pH in chemically assisted fracture, and to a lesser extent the role of ionic concentration are important in stress corrosion cracking. Data on very slow crack growth and the stress corrosion limit are also presented. These data suggest that stress corrosion cracking may play an important role in compound earthquakes and in asperity breakdown in faults. The comminution literature is also reviewed in order to assess the role of chemically assisted comminution in frictional deformation. It appears that chemically assisted comminution may be important at low and high ionic strength because it may reduce the effective viscosity and the shear strength of fault gouge. At intermediate ionic concentration the role of pH, as an agent which enhances crack and fracture propagation, appears to be more important in reducing the coefficient of sliding friction.     

Formation of conical fractures in sedimentary basins: Experiments involving pore fluids and implications for sandstone intrusion mechanisms

Large sand intrusions often exhibit conical morphologies analogous to magmatic intrusions such as saucer-shaped or cup-shaped sills. Whereas some physical processes may be similar, we show with scaled experiments that the formation of conical sand intrusions may be favoured by the pore-pressure gradients prevailing in the host rock before sand injection. Our experiments involve injecting air into a permeable and cohesive analogue material to produce hydraulic fractures while controlling the pore pressure field. We control the state of overpressure in the overburden by applying homogeneous basal pore pressure, and then adding a second local pore pressure field by injecting air via a central injector to initiate hydraulic fractures near the injection point. In experiments involving small vertical effective stresses (small overburden, or high pore fluid overpressure), the fracturing pressure (λ_fract) is supralithostatic and two dipping fractures are initiated at the injection point forming a conical structure. From theoretical considerations, we predict that high values of λ_fract are due to strong cohesion or high pore fluid overpressure distributed in the overburden. Such conditions are favoured by the pore pressure/stress coupling induced by both pore pressure fields. The dips of cones can be accounted for elastic-stress rotation occurring around the source. Contrary to magmatic chamber models, the aqueous fluid overpressure developed in a parent sandbody (and prevailing before the formation of injectites) may diffuse into the surrounding overburden, thus favouring stress rotation and the formation of inclined sheets far from the parent source. For experiments involving higher vertical effective stresses (thick overburden or low pore fluid overpressure), the fracturing pressure is lower than the lithostatic stress, and a single fracture is opened in mode I which then grows vertically. At a critical depth, the fracture separates into two dilatant branches forming a flat cone. We make use of a P.I.V. (Particle Imaging Velocimetry) technique to analyse plastic deformation, showing that these inclined fractures are opened in mixed modes. Close to the surface, they change into steep shear bands where fluids can infiltrate. The final morphology of the fracture network is very similar to the common tripartite architecture of various injection complexes, indicating that different mechanisms may be involved in the formation of dykes. Feeder dykes under the sill zones may open as tensile fractures, while overlying dykes may be guided by the deformation induced by the growth of sills. These deformation conditions may also favour the formation of fluid escape structures and pockmarks.     

Topographic stress and rock fracture: a two‐dimensional numerical model for arbitrary topography and preliminary comparison with borehole observations

Theoretical calculations indicate that elastic stresses induced by surface topography may be large enough in some landscapes to fracture rocks, which in turn could influence slope stability, erosion rates, and bedrock hydrologic properties. These calculations typically have involved idealized topographic profiles, with few direct comparisons of predicted topographic stresses and observed fractures at specific field sites. We use a numerical model to calculate the stresses induced by measured topographic profiles and compare the calculated stress field with fractures observed in shallow boreholes. The model uses a boundary element method to calculate the stress distribution beneath an arbitrary topographic profile in the presence of ambient tectonic stress. When applied to a topographic profile across the Susquehanna Shale Hills Critical Zone Observatory in central Pennsylvania, the model predicts where shear fractures would occur based on a Mohr–Coulomb criterion, with considerable differences in profiles of stresses with depth beneath ridgetops and valley floors. We calculate the minimum cohesion required to prevent shear failure, C_min, as a proxy for the potential for fracturing or reactivation of existing fractures. We compare depth profiles of C_min with structural analyses of image logs from four boreholes located on the valley floor, and find that fracture abundance declines sharply with depth in the uppermost 15 m of the bedrock, consistent with the modeled profile of C_min. In contrast, C_min increases with depth at comparable depths below ridgetops, suggesting that ridgetop fracture abundance patterns may differ if topographic stresses are indeed important. Thus, the present results are consistent with the hypothesis that topography can influence subsurface rock fracture patterns and provide a basis for further observational tests. Copyright © 2014 John Wiley & Sons, Ltd.     

构造型水库地震的断裂力学分析

常见的水库地震有三种：断层破裂型（构造型）、浅表应力局部调整型（微破裂型）和岩溶型，构造型水库地震震级较高。本从地应力、库水压力、断层渗透水压力的分布规律着手，应用弹性断裂力学原理分析了三种应力产生的应力强度因子及总的应力强度因子，并从复合型应力强度因子的断裂判据，分析了水库地震的发震规律。     

Analysis of crack movements observed in an alpine bedrock cliff

Crack widths and rock temperatures were monitored on an andestic bedrock cliff in the summit area of the Daisetsu Mountains, Hokkaido, northern Japan. Sequential data recorded the gradual widening of a crack to the point of critical crack extension, which resulted in catastrophic rock breakage. The data indicate that a combination of liquid water in?ltration into crack tip and subsequent freezing is the most signi?cant factor contributing to critical crack extension. The recorded sub‐critical crack movements involved a number of minor crack extensions and contractions, the timing of which correlates well with the magnitude of the reconstructed thermal stresses at the crack tip derived from thermal deformation of the plate‐shaped rock fragment. Larger crack extensions occurred when stress at the crack tip exceeded a threshold value, possibly re?ecting the control of rock fracture mechanics by which cracks are thought to propagate when the stress intensity factor at the crack tip exceeds the threshold values for stress corrosion cracking and the fracture toughness of the material. Copyright © 2004 John Wiley & Sons, Ltd.